Remark 12 Note that similarly to Remark REF  we get that the conditions are satisfied for all integral bases (with possibly different constants) as soon as they are satisfied for one particular integral basis. Also, the expression in (REF ) is independent of the integral basis for \(U_\infty = \emptyset \)  (as only \(s_\infty \)  depends on \(\mathbb {E}\) ). Note that the dependence on \(\mathbb {E}\)  is not just a technicality, but reflects the fact that the basis alters the way the density of \(U_\infty \)  is measured (as it is a cone and not a lattice, its density is not invariant under \(\mathbb {Z}\) -module isomorphisms). We could easily develop our theorem in the same setting as in [1]}, where the box \(O(H,\mathbb {E})^d\)  is replaced by \(H \Omega _\infty \cap \mathfrak {o}^d\)  for some bounded set \(\Omega _\infty \subset k_\infty ^d\)  with \(\mu _\infty (\partial \Omega _\infty )=0\)  and \(\mu _\infty (\Omega _\infty )>0\) . The proof would be the same and the results would only differ by normalizing constants.
